- Unsolved problems in mathematics
This article lists some

**unsolved problems in**. See individual articles for details and sources.mathematics **Millennium Prize Problems**Of the seven

Millennium Prize Problems set by theClay Mathematics Institute , the six ones yet to be solved are:

* P versus NP

* TheHodge conjecture

* TheRiemann hypothesis

*Yang-Mills existence and mass gap

*Navier-Stokes existence and smoothness

* TheBirch and Swinnerton-Dyer conjecture Only thePoincaré conjecture has been solved. The smooth four dimensionalPoincaré conjecture is still unsolved. That is, can a four dimensional topological sphere have two or more inequivalent smooth structures?**Other still-unsolved problems****Additive number theory***

Goldbach's conjecture and its weak version

* The values of $g(k)$ and $G(k)$ inWaring's problem

*Collatz conjecture ($3n\; +\; 1$ conjecture)

*Gilbreath's conjecture **Number theory: prime numbers***

Catalan's Mersenne conjecture

*Twin prime conjecture

* infinitely manyprime quadruplet s

* infinitely manyMersenne prime s (Lenstra-Pomerance-Wagstaff conjecture ); equivalently, infinitely many evenperfect number s

* infinitely manyregular prime s, is their density $e^\{-^1!/\_2\}$

* infinitely manyCullen prime s

* infinitely manypalindromic prime s in base 10

* infinitely manyFibonacci prime s

* Is everyFermat number composite for $n\; >\; 4$?

* Is 78,557 the lowestSierpinski number ?

* Is 509,203 the lowestRiesel number ?

*Fortune's conjecture (that noFortunate number is composite)

*Polignac's conjecture

*Landau's problems **General number theory***

abc conjecture

* existence of odd perfect numbers

* existence ofquasiperfect number s

* existence of oddweird number s

* existence ofLychrel number s

* Proof that 10 is asolitary number

* existence of Taxicab(5, 2, n) for "n">1.

*Brocard's problem : existence of integers, "n","m", such that "n"!+1="m"^{2}other than "n"=4,5,7**Discrete geometry*** Solving the

Happy Ending problem for arbitrary $n$

* Finding matching upper and lower bounds for K-sets and halving lines

* TheHirsch conjecture on the lengths of shortest paths in the vertices and edges of aconvex polytope **Ramsey theory*** The values of the Ramsey numbers, particularly $R(5,\; 5)$

* The values of theVan der Waerden number s**General algebra***

Hilbert's sixteenth problem

*Hadamard conjecture

* existence of perfect cuboids

* Existence of quadratic number fields being Euclidean but not norm-Euclidean**Combinatorics*** Number of

Magic square s OEIS|id=A006052

* Finding a formula for the probability that two elements chosen at random generate thesymmetric group $S\_n$

* Frankl'sunion-closed sets conjecture that any family of sets closed under unions has an element contained in half or more of the sets

* TheLonely runner conjecture : if $k$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be more than a distance $1/(k+1)$ from each other runner) at some time?**Graph theory***

Erdős-Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs

* The Hadwiger conjecture relating coloring to clique minors

* The Ringel-Kotzig conjecture on graceful labeling of trees

* TheHadwiger–Nelson problem on the chromatic number of unit distance graphs

* Deriving a closed-form expression for thepercolation threshold values, especially $p\_c$ (square site)

* Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without thePetersen graph as a minor has a nowhere-zero 4-flow**Analysis***

Schanuel's conjecture

*Lehmer's conjecture

*Pompeiu problem

* Is $gamma$ (theEuler-Mascheroni constant ) irrational?**Group theory*** Is every finitely presented

periodic group finite?

* Theinverse Galois problem

* For which positive integers "m", "n" is thefree Burnside group nowrap|B("m","n") finite? In particular, is nowrap|B(2, 5) finite?**Other***

Generalized star height problem

*Invariant subspace problem

* Modelingblack hole mergers

*Problems in Latin squares

*Problems in loop theory and quasigroup theory **Problems solved recently***

Road coloring conjecture (Avraham Trahtman , 2007)

* TheAngel problem (Various independent proofs, 2006)

*Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)

*Green–Tao theorem (Terence Tao , 2004)

*Poincaré conjecture (Solution byGrigori Perelman in 2002 now confirmed)

* Catalan's conjecture (Preda Mihăilescu , 2002)

*Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001)

* TheLanglands program for function fields (Laurent Lafforgue , 1999)

* Taniyama-Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)

*Kepler conjecture (Thomas Hales, 1998)

*Milnor conjecture (Vladimir Voevodsky , 1996)

*Fermat's Last Theorem (Andrew Wiles , 1994)

* Bieberbach conjecture (Louis de Branges, 1985)

*Four color theorem (Appel and Haken, 1977)**ee also*** Hilbert's 23 problems

*Timeline of mathematics **References*** [

*http://unsolvedproblems.org/ Unsolved Problems in Number Theory, Logic and Cryptography*]

* [*http://www.claymath.org/millennium/ Clay Institute Millennium Prize*]

* [*http://mathworld.wolfram.com/UnsolvedProblems.html Unsolved problems page at MathWorld*]

* Winkelmann, Jörg, " [*http://www.iecn.u-nancy.fr/~winkelma/mirror/unibas/problem.html Some Mathematical Problems*] ".9 March 2006 .

* [*http://www.geocities.com/ednitou/ List of links to unsolved problems in mathematics, prizes and research.*]**Books discussing unsolved problems***

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***Books discussing recently solved problems***

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***Resources*** [

*http://garden.irmacs.sfu.ca Open Problem Garden*] The collection of open problems in mathematics build on the principle of user editable ("wiki") site

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